FREQUENCY OFFSETS
To increase the channel capacity and reliability of a communication system, the in formation d ata can be transm itted and received by using a number of transmit and receive antennas [67]. This configuration of the communication system is called a MIMO system. Recent research in communication theory [6 8] has demonstrated
th at large gains in capacity and reliability of communications over wireless channels can potentially be achieved by exploiting spatial diversity through MIMO anten nas [69-71]. Spatial diversity can be used to either increase the capacity or enhance coverage in a wireless communication system. In the first case, multiple antennas are used at the transm itter and the receiver to form multiple spatial channels to transmit multiple d ata streams through various spatial modes, which is termed data multiplex ing [31,32,72]. In the second case, multiple antennas are used to transmit multiple copies of the same d ata so th a t a b etter BER performance can be obtained at very low SNR [73,74]. This is exploited to increase the coverage in a wireless commu nication system. Therefore, multiple antennas at both the transm itter and receiver are very likely to play a key role in future high d ata rate wireless communication systems. Often, MIMO transmission schemes proposed in the literature are based
Section 4.1. Problem S ta te m e n t 80
on somewhat idealized assumptions. Such as most MIMO transmission schemes are designed for frequency-flat channels [33,73]. However, if there are multipath signals with large propagation delays, then the assumption of frequency-flat channel might not be valid, depending on the symbol duration.
Besson et al. [33] discussed the estim ation of FOs in MIMO flat fading channels with distinct FOs between each transm itter and receiver. This chapter extend this work for MIMO frequency selective channels th a t allows distinct FOs for each multipath between each transm it and receive antenna. As discussed in chapter 1, the perfor
mance of such multiple antenna based systems may seriously degrade in the presence of FOs. Therefore, it is of importance to determine these FOs and to take them into account for in the equalizer design. In this scenario, to estim ate the FOs and MGs, an AML estimator is proposed th a t exploits the correlation property of the transmitted training sequence. In order to assess the performance of the proposed estimator the corresponding CRLB is determined, and used as a benchmark for the performance of the proposed estimator. Furthermore, multiple FOs introduce deterministic time variations in the channel, which are exploited to design a low complexity MIMO recursive MMSE equalizer to account for symbol-by-symbol variations in the COM.
4.1 Problem Statem ent
Consider a MIMO communication system with tit transm it and hr receive antennas, where the signal between any two transm it and receive antennas has propagated through a total of L different paths, with each path possibly having different FO. If the sampling rate at the receive antenna k is equal to the symbol transmission rate then the received baseband signal can be w ritten as
Section 4.1. Problem S ta te m e n t 81 where tit L — 1 uk{m) =
E E
hki(p)ejuJklpinxi(m - p ), 1 = 1 p = 0 (4.1.2)for m = n , . . . , n — N + 1 , k = 1 , . . . , n/*, where N is the number of symbols received,
hki(p) and ujkip are respectively the MGs and FOs between the receive antenna k and the transm it antenna /, for the m ultipath p ; these are assumed to be quasi-stationary, i.e., they do not change significantly over an observed d ata frame, but may change be tween the frames. Here, {x/(m)} is the training signal sequence, transm itted from the Ith transm it antenna and Vk(m) is assumed an additive zero mean circularly Gaussian distributed, spatially and temporally uncorrelated, white noise with variance a Let X/p = diag | x/p j , where diag{q} denotes a diagonal m atrix with the vector q along the diagonal, and
I? xi(n — p) ••• xi(n — p — N + 1) X/p ew(p) = h ki = eJUklpn hki( 0) e j u ki p { n - N + l ) hki(L — 1) (4.1.3) (4.1.4) (4.1.5) where the vector e u(p) contains the FO between the receive antenna k and the trans mit antenna I, for the m ultipath p and h ki is the vector of MGs between the receive antenna k and the transmit antenna /. Furthermore, suppose th at
V « = h k = V* = X/oejt/(0) h H hL Xf(£.-i)e,ti(L — 1) . T h L T N x L titL x 1 N x rixL (4.1.6) (4.1.7) (4.1.8) "Vfcl "V k2 * "V kriT
Using these identities, the N received samples in (4.1.1) can be written in vector form as
T
r k = rk{n) ••• rk( 7 i - N + l )
Section 4.2. E stim ation o f M ultipath Gains and Frequency Offsets 82
To estimate the various MGs and the FOs, suppose
and Uki = Uk = WhlO Ukl 1 U k l ( L - l ) T U>kl LJk2 <jJkriT (4.1.10) n TL x 1 (4.1.11)
then, the unknown param eter vector, 0 k, corresponding to receive antenna fc, to be
T
estimated can be written as Ok = estimating Ok is considered.
h i uj In the next section, the problem of
4.2 Estimation of Multipath Gains and Frequency Offsets
In this section, an approximate maximum likelihood (AML) estimator is outlined, which fully exploits the structure of the transm itted training sequence. Since the noise, u*(n), at the receive antenna is spatially uncorrelated, the parameters asso ciated with each receiver can be estim ated independently from the received signal. Considering (4.1.9), the likelihood function of r*, can be written as [30]
1 ( r f c - V f c h fc) t f ( r f c- V fch fc)
P(r*l #*) = 7Z „ 2 \n e °v ’ (4 -2 -1)
{ w z r
Taking the natural logarithm and ignoring the constant terms, as they will not affect the maximization of the likelihood function, (4.2.1) can be written as
lnp(r*| 0 k) = — (rk - V k h k)H (rk - V k h k) . (4.2.2) In order to estimate parameter vector, 0 k, maximization of (4.2.1) is equivalent to the minimization of (4.2.2) (as the minus sign is omitted in the log-likelihood function) with respect to h*. and yields
h fc = ( V " V t ) -1V " r lfe, (4.2.3)
then, inserting h k into (4.2.2) yields the cost function to minimize with respect to the FOs
Section 4.2. E stim ation o f M u ltip ath Gains and Frequency O ffsets 83
Note th a t minimizing (4.2.4) requires an titL-dimensional minimization. However,
choosing the training sequence {x/(n)} such th at
E {x*(n - u )x d(n - r/)} = 8u- v8i-d, (4.2.5) this minimization can be decoupled into one-dimensional minimizations. Con sidering (??), note th a t will be dominated by the large diagonal terms, with al most negligible contribution from the off-diagonal terms. Thus, lxKn )|21 /cl, where /c is constant over the frame considered, enabling us to approximate the minimum of (4.2.4) as the maximum of
J \ uj klp) = i f V fcV " r fc tit L — 1
= £ £
1 = 1 p= o N - l ^ r*k{n)xi{n — p)eju}klpTl n = o (4.2.6)Maximizing J'{ujkip) is equivalent to maximizing all individual terms of the above outer sum. Consider a given path s from the transm it antenna q to the receive antenna k, the contribution of this path to the cost function, J'(ujkip), can be written as
Tpkqs(n) = r*k(n)xq{n - s)eiWfc*'n
and using (4.1.1) 'fikqsin) = h*kq(s) \x q(n ~ 5 ) | 2 + ^i(n) + c2(n), (4.2.7)
where C i ( n ) = vk{n)xq{n — s)ePu>kqan rix L — 1
C2{n) = ^2 ^ hkl(P)XKn - p ) Xq(n - S)e3AUJklpn>
1= 1 P= o pjts | l=qand AuJkip = ukqs — <^kiP- The first term in 'ipkqs(n) corresponds to the signal power for the path s, and as E{x*(n — u)xd(n — v)} = 0 for u ^ v, or I ^ d, it will be significantly larger than the contribution from the interference term, c2(n), constructed from all
Section 4.3. N um erical E xam ple for th e V ariance o f th e E stim ators 84
paths except p ^ s when transm it antenna I = q. Thus, each of the terms in the outer sum of (4.2.6) will be maximized for u = Ukqs, suggesting the AML estimator for ujkqs, for q = 1 , . . . , n T, and s = 0 , . . . , L — 1, as
2 N - l kin )xq(n - s ) ^ n = 0 ^ k qs — &rg max <pkqs (k-0 ) (4.2.8) (4.2.9) which can be efficiently evaluated using the FFT. Once the FOs axe estimated, the MGs, hfc, can be estim ated by inserting the estimated values of the FOs in (4.2.3). It can be noted th a t the AML for FOs does not provide the block based solution but for MGs, the AML solution is block based.
4.3 Numerical Example for th e Variance of the Estimators
To illustrate the theoretical findings a numerical example is provided. Consider a case with maximum vehicular speed of 250km/h, at a carrier frequency of 1800MHz (RA250 channels as defined in GSM standard [43]), this corresponds to the maxi mum DS of 0.005 when normalized to the symbol rate of lOOkbits/sec, which will be different for different angles of arrivals. For higher carrier frequency or speed, the normalized frequency will further increase. In order to see the performance of the proposed estim ator, a case using two transm it and two receive antennas is simulated. Here, it is considered th a t there are two paths between each transm it and receive an tenna, allowing eight different paths and correspondingly eight different FOs. Using the assumption of a quasi-stationary channel, the channel parameters, A^(p), and the FOs, ujkip, remain constant throughout the training burst interval, but may change according to the Rayleigh distribution between the bursts. BPSK signals are used for training data, the length of the training data from each antenna is 200 samples. The training signals transm itted from each antenna are assumed uncorrelated. In order to estim ate the FOs, an FFT based method is used, therefore, the difference between
Section 4.3. N um erical Exam ple for th e V ariance o f th e E stim ators 85
any two FOs is assumed to be greater than 1/N. In the simulation, parameters are estimated and the variances of the estimators are compared with the corresponding CRLB, which is derived in Appendix 4A. Figures 4.1 and 4.2 depict the variance of the estimators for the MGs and the FOs, respectively. The simulation results show th at the proposed estim ator attains the CRLB. It can be noticed th at the variance of the estimation error sometimes goes slightly below the CRLB. The reason for this can be attributed to the fact th a t an FF T based grid search method is used to estimate the FOs. The performance of this method relies on the chosen resolution of the FFT and hence the estimate of the FO. Therefore, for the variance of the estimation error to match the theoretical CRLB, the resolution should be infinitely small so that the frequency estim ate is unbiased. As a consequence of the non-ideal resolution and hence the bias in the frequency estim ate the error variance may sometimes go a little below the theoretical CRLB which assumes exact frequency knowledge.
Section 4.3. Numerical Example for th e V ariance o f th e Estim ators 86
19
SNR
F ig u re 4.1. Comparison of the variance of the estimates of channel gains (dashed line) with the corresponding CRLB (solid line).
O H | • 19- 0 1 T T T 13 SNR
F ig u re 4.2. Comparison of the variance of the estimates of FOs (dashed line) with the corresponding CRLB (solid line).
Section 4.4. A MIMO R ecursive M M SE Equalizer Design 87
4.4 A MIMO Recursive MMSE Equalizer Design
In order to design a recursive MIMO equalizer for the transm itted symbols from antenna /, the equalizer baseband model given in Figure 4.3 is used.
F ig u re 4.3. A two transm itter and three receiver MIMO transm itter and receiver baseband system.
From the figure, the equalizer output for the symbol transm itted from antenna I can be written as
/
yi(n) = w i i w /2 wln R r l m **2 m \ (4.4.1) \ r n Rm ) wherew lk wik(°) wik W wik(M ~ !)
is the length M equalizer coefficient vector to decode xi(n) from the received samples, , at the antenna k. By rearranging rk{n) rk(n — 1) ••• rk(n - M + 1)
the individual equalizer coefficients and received samples in (4.4.1), the equalizer output can be w ritten as
yt{n) = w //r m
= w /yH cx m + w //v m,
(4.4.2) (4.4.3)
Section 4.4. A MIMO Recursive M M SE Equalizer Design 88 where w = w (0 w (0)r w ( l ) r u'll(i) w u (l) w (M — 1)' T U>l nR ( l ) , 1 x n RM l x % rm = r T(n) • • • r T(n - M + 1) and r(n) - n W rnR(n) . More-
over, H c is the n RM x n T(M + L — 1) CCM and x m is the nT(M + L — 1) x 1 transm itted signal vector and are defined as
H , = H n = K = H" K (p) hy h j h j(0 )r h j (l) hkl( p ) e ^ r " hnn R ■ K ( L - 1 ) T hknT (p)eiUknTPn x T(n) x T(n — M — L + 2) x(n) = Finally, v m = v T(n) • • xi (n) . . . x nT{n) 1 T v T(n — M + 1) and v (n) = v\ (n) V n R ( n )
R em ark 1. Here, to estimate the transm itted symbols the condition t i r M >
t i t { M + L — 1) must be satisfied or
L - 1 nR ^ n T 1 +
M (4.4.4)
which implies th at in a m ultipath channel n R > nr- This result contrasts with the result mentioned in [69] th a t says n R = tit• Moreover, M > (^~ I nT:(L — 1)^.
The length M equalizer is obtained by minimizing the mean square error cost function J = E{\yt(n) - xi(n - d)|2}, (4.4.5)
Section 4.4. A MIMO Recursive M M SE E qualizer Design 89
where I e ( 1 , 2 , n T) and d e (0 ,1 , M + L — 2). Therefore,
w = (4.4.6)
where z v is the n r ( M + L — 1) x 1 coordinate vector, only containing a non-zero
element at position i>, i.e.,
The position of the non-zero element in z v determines the equalizer corresponding to the various transm itters I 6 (1 ,2 ,..., nT) and retrieval delays d 6 (0,1,..., M + L — 2).
For example the one at position n r d + 1 will design an equalizer to decode the trans-
to the signal transm itted from antenna I can be found by just multiplying with the corresponding z v. Due to FOs, the COM H c changes after every symbol. Therefore, it is necessary to update the equalizer coefficient values at every symbol, which is typically computationally infeasible. To deal with this problem, exploiting the struc tural movement of the submatrices in R a computationally efficient recursive scheme is proposed. This is an extension of the single user result presented in the previous chapter to the multiuser system considered herein. To emphasize the fact that R changes at every symbol time n, subscript n in (4.4.6), is used as follows
(4.4.7)
mitted signal from antenna I with delay d. The derivation of the equalizer coefficient vector is given in Appendix 4B.
Once (H CH ^ + ^ I ) -1H C is known, the equalizer coefficient values corresponding
w n = R n1H c(n)zv (4.4.8)
and
w n+i = R ni1H c(n + l)z w (4.4.9)
A close inspection of H c(n) reveals th a t the matrices R„ and Rn+i can be written as follows
Section 4.4. A MIMO Recursive M M SE E qualizer Design 90 Rn = R - n + 1
....
1 0 3 c n I ... O 1 G 3 E„ _ E " --- 1 e 0 (4.4.10) (4.4.11) where G n e Cn«(M" 1)xn«{M_1), B n 6 CnRXnR, D n € Cn*xn«, C n e CnR{M~l)xnR,and E n E CnRXnR^M~l\ Note how the Hermitian matrix G„ moves from the top left corner to the bottom right corner from time n to n + 1. Further, if the inverse of G n, is known, then one could find the inverses of R „ and Rn+i using the matrix inversion lemma (see, e.g., [53]), yielding a computationally efficient update of w. As G n will not appear in Rn+2, it can not be used to find the inverse of Rn+2- Thus, the
scheme so far only allows for a pairwise computational saving, still requiring inver sion of G n+i, to find the inverse of R „ + 2 efficiently. However, further exploiting the
structure, one may compute the inverse of G„+i efficiently from the inverse of Rn+i using the following lemma obtained in the previous chapter.
Let Q l l Q l 2 Q 2 1 Q 2 2 - 1 H n H12 H 2i H2 2 (4.4.12)
Here, dim {H ^} = dim{Qfc/}, where dim{.} denotes the dimension of matrix. Then, provided the relevant inverses exist, the inverse of matrix Q n can be written as the Schur complement of H2 2, i.e.,
Q n1 = H „ - H i2H2-2‘H21 (4.4.13)
Therefore, starting at time n, the inverse of the sub-matrix G„ is found. The inverses of R „ and Rn+i are then found using the matrix inversion lemma. Once the inverse
Section 4.5. Sim ulation 91
of Rn+i is known, the inverse of G n+i is found using (4.4.13). Further, the inverse of G n+i can then be used to find the inverse of Rn+2, and so on. This is called a
forward and backward recursion method to find the inverse of the matrix R „ at every symbol time n. Thus, the explicit inverse of the sub-matrix G n is needed only once at the start; thereafter, only the inverse of H2 2 is required after every symbol, which
significantly reduces the complexity of finding the inverse of R„ from 0 ( n 3R M 3) to 0 ( n 3R).
R e m a rk 2. If only one path exists between any two transm it and receive antennas, then the m atrix H c(n )H ^ (n ) will be block diagonal, enabling the inverse to be found by taking the inverse of individual blocks in H c(n )H ^ (n ).
R e m a rk 3. For single transm it and single receive antenna schemes with distinct FOs for each path, H2 2 is only a scalar as shown in Chapter 3. Hence, for this case, the
proposed recursive m ethod does not require any explicit m atrix inversion, whereas the conventional methods require inversion of an M x M matrix at every symbol.
4.5 Simulation
In order to dem onstrate the benefits of employing FOs in equalization, a MIMO channel is considered. In this simulation a 2 transm it and 3 receive antenna system is considered. The number of m ultipaths between each transm it and receive antenna is assumed equal to 2. The equalizer is designed using 4 taps. The FOs are initially
chosen to be of the order of 1 0~2, but are changed at every burst according to a
random walk model f k ( n ) = fk(n — l) + 0.0 0 1u(n), where u{n) is the Gaussian random
variable with zero mean and variance equal to 1. For simulations, three scenarios are considered. In the first scenario, FOs associated with each multipath are set to zero and the design equalizer is based on the MMSE criterion. In the second scenario, distinct FOs are considered from each m ultipath and are changed according to a random walk model after each frame. To compensate for the effects of multiple FOs, the proposed recursive equalizer is used th at account for FOs in equalizer design. In
Section 4.5. Sim ulation 92
the third scenario, a channel with FOs as in the second scenario is considered but the designed MMSE equalizer ignores the effects of FOs in the equalizer design. The bottom curve depicted in Figure 4.4 shows the benchmark performance in the first scenario, while the middle curve shows the performance of the proposed algorithm in the second scenario and finally the top curve shows the performance of an equalizer th at does not accounts for FOs. The performance of the proposed recursive equalizer is close to the bench mark performance and in the third scenario the performance of the equalizer not accounting FOs is independent of SNR.
2
0
1
- 0 - Frequency offsets with our proposed correction No frequency offsets
Frequency offsets without correction ___ SNR
F ig u re 4.4. Bit error rate performance comparison of the proposed scheme account ing for FOs in the equalizer design with the conventional equalizer scheme ignoring the FOs in the equalizer design. For bench mark the simulation result of a conventional scheme when there is no FO in the channel is also shown.
Section 4.6. S um m ary 93
4.6 Summary
In this chapter, the estimation problem of the MGs and FOs for the frequency selec tive MIMO channel with distinct FO was addressed. By exploiting the correlation property of the transm itted pilot signal an AML estim ator was proposed that de composed the tltL dimensional FOs estimation maximization problem into one dimensional FO estimation maximization problems. The performances of the esti mators were validated by comparing their variances with the corresponding CRLB, which was also derived. The estim ators were found to be both computationally and statistically efficient. Then, the structural movements of the matrices inside the big m atrix R „ was dem onstrated at every symbol and a recursive equalizer was proposed th a t reduced the computational complexity significantly. Finally, simulation results showed substantial improvement in the performance, when the FOs were considered in the equalizer design, as opposed to an equalization w ithout consideration of FOs.
Section 4.7. A ppendices 4 94
4.7 Appendices 4
Appendix 4A: Derivation of Cram er Rao Lower Bound for MIMO
This section is devoted to the derivation of the Cramer Rao lower bound for the estimators of MGs and FOs. Stacking all the received samples from time n to (n — N + 1), from all antennas, (4.1.1) can be w ritten in vector form as
r = u + v, (4.7.1) where r = r(n ) = r (n)T r\(n) r(n — TV + 1): 1 T rnR(n)
with u and v formed similarly. Denote the unknown desired vector parameters
where r) A Vk = T T m m T (4.7.2) (4.7.3) R e ( h k)T I m ( h k)T u>l
Since the noise sequence Vk(n) is spatially uncorrelated, the Fisher Information Matrix (FIM) for the estim ation of 77 can be found using Slepian-Bangs formula (see, e.g, [30],
[53]). 2 / d u H du F (k,l) = —-zRe ST* v n = 0 V dVk dViT d\iH(n) du(n) dr)k dr)? (4.7.4) where <9uH drfk du d r ? d u 8 R e ( h k ) d u H d l m ( h k ) d u H duik (3titL x n RN) d u d u d u d R e ( h ,r ) d l m ( h i r ) du>,T (nRN x 3titL )
Section 4.7. A ppendices 4 95
Here, k ,l = 1,2, ...,7 7/*. The FIM can be w ritten as
F =
F ( l, 1) F ( l, 2) ••• F ( l,n * )
F ( 2 ,1) F ( 2 ,2) ••• F(2 , n R)
(4.7.5) F ( n R, l ) F(rift,2) ••• F {nR, n R)
where F(A:, I) denotes the (A;, /)th sub-m atrix of the FIM corresponding to the parame ters rjk and r^. From (4.7.4), it can be noted th a t F(A;, I) = 0 whenever k ^ I. Hence, there is a decoupling between the estim ation error in parameters corresponding to two different receive antennas and the FIM is block diagonal, which justifies th a t the parameters corresponding to each receive antenna can be estimated independently. Let F t = F ( k, k ), the FIM of size 3t i t L x StitL corresponding to the estimation of
rjk = [Re(hk)T I m ( h k)T w*]7’, then F* can be represented as
F k[Re(hk) , R e ( h k)] F fc[i?e(hfc), I m ( h k)} F fc[/?e(hfc), u k\
F fc = ^ F * [/m (h fc) ,/t e ( h fc)] F * [/m (h * ),/m (h* )] F k[Im(hk),Ljk] (4.7.6) F k[u>k, R e ( h k)] F* [<*>*, I m ( h k)] F ^ u ^ u ;* ]
and the elements of F^ can be found using the differentials
d Re h kl(p) duk(n) - p) d l m h ki(p) duk(n) = j n h kl{ p ) e ^ nXl{ n - p ) OUlklp (4.7.7b) (4.7.7a) (4.7.7c)
Section 4.7. A ppendices 4 96 Introduce U* = P ? P *
Pfc =
[p*i(0) Pf c i ( L- l )•••
pfcnr( L - l ) ] p k i ( p ) = X/pe/tz(p) D n = diag (0 ,1, • • • , TV — 1) D h = diag (hkl { 0), • • • hkl(L - 1), • • • , hknr(L - 1)) T t = P " D „ P fcD A S* = D " P " D j P t D t B = [ f i e f S t - T f U ^ T t) ] - 1The individual elements corresponding to the estimation of rjk can be found from (4.7.4). Therefore, the initial row of the submatrices in (4.7.6) can be written as
F t [fle(h t), fle(h t)]
=
Re [P ?P * ] (4.7.8) F t [« e (h t), /m (h t)]=
- I m [P ?P * ] (4.7.9) Ffc[/fe(h*),a>*]=
- I m [P ? D nP t D t] (4.7.10) The second row of matrices can be w ritten asF t (/m (h t), fle(h t)]
=
Im [P ?P * ] (4.7.11) F t [Im(h*), /m (h t)]=
Re [ P f P*] (4.7.12)Ft
[/m (h*),wt] =
Re[ P j ^ P t D t ]
(4.7.13) Similarly, the third row of matrices can be w ritten asF t
[«*,
fle(h t)]=
- I m [ P f D n P t D t ] " (4.7.14) F t [wt, /m (h t)] = Re [ P ? D nP * D * ]" (4.7.15)Section 4.7. A ppendices 4 97
F* [«*,«*] = fle [D " P " D 2 p )cD fc]
In compact form (4.7.6) can be w ritten as
(4.7.16) F k = — Re{ U fc) -7m (U fc) - I m ( T k) I m ( U k) Re( U fc) R e(T fc) - / m ( T * ) r R e ( T k)T R e (S k) (4.7.17)
Note th a t there is a coupling in the estim ation error between the channel parameters